Exploitation, Exploration and Innovation in a Model of Endogenous Growth with Locally Interacting Agents

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2 Exploitation, Exploration and Innovation in a Model of Endogenous Growth with Locally Interacting Agents Giorgio Fagiolo Giovanni Dosi January 22, 2003 Abstract The paper presents a model of endogenous growth in which firms are modeled as boundedly-rational, locally interacting, agents. Firms produce a homogeneous good employing technologies located in an open-ended technological space and are allowed to either imitate existing similar practices or to locally explore the technological space to find new, more productive, techniques. We first identify sufficient conditions for the emergence of empirically plausible GDP time-series characterized by self-sustained growth. Then, we study the trade-off between individual rationality and collective outcomes by providing an example in which more rational agents systematically perform worse than less rational ones. Keywords: Innovation, Endogenous Growth, Local Interactions, Exploration vs. Exploitation JEL Classification: O30, O31, C15, C22 Thanks to Jesse Benhabib, Alan Kirman, Karl Schlag, Willy Semmler, a few participants to the 6th WEHIA Workshop (Maastricht, June 7-9, 2001) and to the 7th International Conference of the Society for Computational Economics Conference (Yale University, New Haven, June 28-29, 2001), and two anonymous referees for their helpful comments. Sant Anna School of Advanced Studies, Laboratory of Economics and Management (LEM), PISA (Italy). fagiolo@sssup.it. Sant Anna School of Advanced Studies, Laboratory of Economics and Management (LEM), P.zza Martiri della Libertà, 33, I PISA (Italy). gdosi@sssup.it. Tel: Fax:

3 1 Introduction The analysis of the determinants of self-sustained processes of economic growth fueled by technological advances has received an increasing attention in the past few years. On the theoretical side, Endogenous Growth and Evolutionary models have been trying to explain how positive feedbacks in knowledge accumulation affect per-capita income growth (Romer (1990), Grossman & Helpman (1991), Nelson & Winter (1982), Verspagen (1993) and Silverberg & Verspagen (1994)). On the empirical side, a rapidly expanding literature on the economics of technological change has been instead exploring the drivers of innovation and diffusion at the levels of firms, sectors and whole Countries (see, among others, Freeman (1994), Rosenberg (1994), Nelson (1995) and Stoneman (1995)). Notwithstanding this great effort, many scholars have recently spelled out a negative assessment on the extent to which neoclassical, endogenous and evolutionary growth theories have been able to match old and new growth stylized facts and to provide fresh testable implications (cf. Durlauf & Quah (1998), McGrattan & Schmitz (1998) and Silverberg & Verspagen (1996)). As argued by a large body of literature (cf. e.g. Nelson (1998) and Dosi, Freeman & Fabiani (1994)), these difficulties are mainly due to the large gap still existing between what we historically know about the microeconomics of technical change, innovation and technological diffusion, and the ways we represent that knowledge in formal models. For example, economic growth models do not usually account for both systematic heterogeneity observed in technological competencies and the fine details of the mechanisms governing the dynamics of interactions among economic agents. However, microeconomic diversity and institutional settings have been shown to affect in non trivial ways the properties of aggregate dynamics. Hence, any representative agent reduction employed by a good deal of contemporary literature might turn out to be misleading whenever heterogeneity and interactions are important factors in explaining economic growth (see Kirman (1992) and Kirman (1998)). Furthermore, technological advances typically involve business firms whose R&D activity is characterized by routinized decisions, trial-and-error, mistakes and unexpected discoveries (cf. Dosi & Lovallo (1998)). Consequently, forward-looking rationality typically imputed to agents in standard models of growth might not be a good proxy, especially when firms face complicated environments where novelty endogenously emerges as the outcome of others behaviors (cf. Conlisk (1996) and Dosi, Marengo & Fagiolo (2002)). In economies populated by heterogeneous agents (e.g. firms) who repeatedly interact, innovate and adaptively learn about the world where they live in, observed aggregate regularities can hardly be understood as equilibrium paths (Silverberg & Verspagen (1997)). 2

4 Empirically observed properties of macroeconomic time-series might be instead more fruitfully interpreted as metastable regularities emerging in a complex evolving system. For example, the observed regularities displayed by the patterns of self-sustained GDP aggregate growth may be described as emergent properties of an economy composed of many heterogeneous simple firms interacting in some properly defined technological space (cf. Lane (1993a) andlane(1993b)). Following this intuition, we present a computer-simulated model of endogenous growth in which simple, boundedly-rational firms produce a homogeneous good employing technologies located in an open-ended (i.e. without boundaries) productivity space. Technologies located close to each other have similar productivities, while more distant technologies perform better on average. Entrepreneurs can either imitate existing practices (similar to the one they currently master) or locally explore the technological space to find new and more productive techniques (i.e. innovate). We ask whether (and, if so, in which technological regimes) such an economy is able to generate self-sustaining patterns of aggregate growth with statistical properties similar to those displayed by empirically observed time-series. The paper is organized as follows. In Section 2, we outline in more detail the building blocks and theoretical conjectures supporting the model described in Section 3. Next, in Sections 4 through 6, we present an extensive analysis of computer simulations. Section 7 discusses some econometric properties of the simulated time-series. The tension arising in the model between individual rationality and collective performance is illustrated with a simple example in Section 8. Finally, Section 9 draws some conclusions and flags research developments ahead. 2 Decentralized Knowledge Accumulation, Interactions and Collective Outcomes A large body of empirically-grounded contributions has recently investigated the main properties of the processes underlying the emergence of self-sustaining growth patterns. In a nutshell, two key sets of insights emerge from this literature (cf. Rosenberg (1982), Freeman (1982) and Dosi (1982)). First, technological search and knowledge diffusioninpresenceofdynamicincreasing returns seem to play a primary role among the engines of growth. Technological advances are endogenously generated through resource-expensive search undertaken by a multiplicity of agents. Search is generally characterized by radical uncertainty and innovative entrepreneurs are driven by the belief that there might be something profitable out there. 3

5 As agents are generally unable to form probability distributions on the outcomes of their search efforts, systematic mistakes in innovative search and adoption are very likely. Second, the process of technical change appears to be driven not only by innovation but also by time-consuming diffusion (see also Jovanovic & Rob (1989) and Jovanovic (1997)). Innovations are indeed not entirely appropriable and knowledge progressively spreads (with some time lags involved) to other agents who might catch-up by investing in imitation. Knowledge accumulation generally entails dynamic increasing returns at the level of individual agents. However, radically new technologies typically involve discontinuities and only part of the old knowledge might be useful in the exploitation of subsequent technologies. In order to embody the foregoing properties in the present analysis, we will start by describing search and innovation activities, technological diffusion and knowledge accumulation as direct interaction processes taking place in some (high-dimensional) technological space (see Kirman (1998) and Chiaromonte & Dosi (1993)). Suppose indeed that the technologies currently adopted by all firms in the economy, as well as those still to be discovered, are associated to points of a metric space (e.g. a 2-dimensional regular lattice). Any metrics the space is endowed with will then metaphorically represent technological dissimilarity: similar technologies will lie close to each other, while more productive technologies will be situated far from existing ones. Both imitative and innovative activity might be therefore represented as an interaction process in which the sets of interacting units are firms and technologies. Any firm will directly affect the behaviors of other firms employing similar technologies. Since adopted technologies will typically change in time, interaction structures (e.g. who interacts with whom) are likely to endogenously change over time as well. More precisely, we will employ the following metaphor. Think of the technological space as an empty, unbounded sea. The notional production possibility set is composed of a discrete, countable set of production activities (technologies, paradigms, etc.), each of which can be thought as an island randomly placed in a point of the sea and endowed with a mine. The economy is populated by a discrete, finite population of firms (or Schumpeterian entrepreneurs) initially distributed across a small subset of islands (i.e. the set of fundamentals). We assume that an island can be at any point in time exploited by more than one agent, while each agent cannot exploit more than an island at the same time. Every agent currently living on an island represents one of the adopters of technology embodied in it (or, in our metaphor, a miner ) and extracts a homogeneous good (i.e. GDP). Mining is possibly characterized by increasing returns to scale in the number of current adopters due to knowledge-accumulation. Since distances between islands in the sea are a proxy of technological (productivity) differences and the sea is unbounded, notionally 4

6 unlimited opportunities do exist in the economy, albeit at each point in time only a small subset of mines are known and exploitable, i.e. those which have been operated by any one firm so far. We suppose that miners might become explorers by leaving the island they are working on and traveling around to find still unknown, possibly better, islands. The set of fundamentals can be therefore enlarged through endogenous innovations. Alternatively, miners might try to capture informative signals about the location of already known, better, islands and decide whether to imitate firms currently working on them. Of course, this representation of the space of technological opportunities and of the related innovation processes is much more abstract than any particular empirical example. However, we believe that it captures some of their general features, including the intrinsic uncertainty of search, the idiosyncratic and cumulative nature of technological learning, and the painstaking patterns of melioration and diffusion of specific bodies of knowledge (i.e. of technological paradigms). Within this framework, we will address the following issues. First, we will attempt to identify conditions under which the economy is able to tackle the trade-off between exploitation of existing technologies and exploration of potentially superior ones and to generate patterns of self-sustaining growth 1. Second, extensive Montecarlo simulations will be performed so as to map technological regimes (i.e. strength of path-dependency in learning achievements, levels of individual willingness to explore, etc.) into aggregate growth patterns. Third, as a plausibility check, we shall investigate whether the model is able to generate GDP time-series displaying statistical properties similar to the empirically observed ones. Finally, we will discuss the conflict arising in the model between individual rationality and collective economic performance. In particular, we shall investigate a simple situation wherein boundedly rational firms are replaced by a representative agent with unbounded computational skills and complete information about the structure of the economy. 3 The Model Consider a technological space represented as a 2-dimensional, infinite, regular lattice endowed with the Manhattan metrics d 2 1. Time is discrete and the generic time-period is denoted by t =0, 1, 2,... The economy is populated by a finite set of agents I = {1, 2,...,N}, 1 The exploitation-exploration trade-off in individual decisions (as well as its consequences for aggregate efficiency) is studied in March (1991). 2 The distance between any two nodes (x, y) and (x 0,y 0 ) in the lattice is thus: d 1 = x x 0 + y y 0. The choices of the lattice (and its dimension), as well as that of the metrics, do not crucially affect our results. 5

7 N, and a countable infinite number of islands, indexed by j {1, 2,...}. There is only one good (GDP), which can be extracted from any island. Each node (x, y) inthelatticecanbeeitheranislandornotandeachislandhasasize of one node. Let p(x, y) be the probability that the node (x, y) is an island. We will assume that p(x, y) =π, all(x, y), whereπ (0, 1). Eachislandj is completely characterized by its coordinates (x j,y j ) in the lattice and by a productivity coefficient s j = s(x j,y j ) < + (i.e. the amount of good which can be extracted if there is only one agent on j). Each agent i I is in turn characterized, at each t, byherstatea i,t and her position in the lattice (x i,t,y i,t ). The state of an agent a i,t can be: miner, explorer or imitator, i.e. a i,t { mi, ex, im }. Denote by m t (x j,y j ) the number of agents currently working on island j and define an island j to be currently known if m τ (x j,y j ) > 0 for at least a τ :0 τ t, i.e. if it currently hosts some agents or if did host some miners in the past. Accordingly, let the set of currently known islands be defined as: L t = {j =1, 2,...: 0 τ t : m τ (x j,y j ) > 0} (1) Let us call colonized a known island which is currently exploited at t, i.e. an island j L t : m t (x j,y j ) > 0. Conversely, all islands which are not in L t will be unknown, since no agent has previously exploited them. Finally, denote the cardinality of L t by l t.letus turn now to describe how the economy evolves. 3.1 Production Suppose that at time t agent i I is a miner currently located on island j L t with coordinates (x j,y j ).Weassumethati will extract, at no cost, an output Q i,t given by: Q i,t = s(x j,y j )[m t (x j,y j )] α 1 (2) where α 1. Hence, the current total output of island j L t will be: Q t (x j,y j )=s(x j,y j )[m t (x j,y j )] α. (3) Total output (GDP) will obviously read: Q t = P j L t Q t (x j,y j ). 3.2 Exploration and Innovation At time t, each miner currently working on island j L t decides to become explorer (i.e. a i,t+1 = ex ) with probability ɛ 0, whereɛ is taken to measure the willingness to explore 6

8 of agent i (which in this first approximation is the same for all agents). If i decides to become explorer, she leaves island j and sails around until another possibly still unknown island is discovered. During the search, explorer i is not able to extract any output and moves through the lattice following the naïve stochastic rule: Prob{(x i,t+1,y i,t+1 )=(x, y) (x i,t,y i,t )} = 1 4 x x i,t + y y i,t =1, all (x, y) (4) While exploring, each agent carries the memory of the last quantity of output produced in the state of miner that is Q i,τ,whereτ is the last period of mining before leaving. The new location of the explorer (x i,t+1,y i,t+1 ) might obviously be: (i) sea ; (ii) a known island j L t ; (iii) a new island j {1, 2,...}\L t. Let us focus on the third case 3. If the node inspected by explorer i at time t +1is a new island (which happens with probability π),weassumethatthenewislandj with coordinates (x j,y j )=(x i,t+1,y i,t+1 ) is added to the set of known islands, i.e. L t+1 =L t {j } and l t+1 = l t +1. In order to capture the crucial distinction between innovations within existing knowledge bases and introduction of radically new technological paradigms (cf. Dosi (1982)), we let the intrinsic productivity coefficient of a new island j discovered by an explorer carrying the output memory Q i,τ to be: s j = s(x j,y j )=(1+W ) {[ x j + y j ]+ϕ Q i,τ + ϖ}, (5) where W is a Poisson-distributed r.v. with mean λ > 0, ϖ is a uniformly-distributed r.v., independent of W, withmeanzeroandvariance1; and, finally, ϕ [0, 1]. The interpretation of Eq. (5) is straightforward. The initial productivity of a new island depends on four factors, namely: (i) its distance from the origin; (ii) past skills of the discoverer, i.e. ϕ Q i,τ (that is, a cumulative learning effect); (iii) a random variable W which allows for low probability high jumps (i.e. changes in technological paradigms); (iv) a stochastic i.i.d. zero-mean noise ϖ controlling for high-probability low-jumps (i.e. incremental innovations). 3.3 Interactions, DiffusionofKnowledgeandImitation Exploitation of existing technologies is not associated to production only. Indeed, miners might also decide to imitate currently known technologies by taking advantage of infor- 3 In the first case (x i,t+1,y i,t+1 ) 6= (x j,y j ) for all j, anda i,t+1 = ex (i.e. the exploration goes on), while in the second case, there will be a j L t such that (x i,t+1,y i,t+1 )=(x j,y j ) and hence the explorer i becomes miner on j L t, i.e. a i,t+1 = mi. 7

9 mational spill-overs emanated by more productive islands located in their technological neighborhood. More formally, the process of knowledge diffusion and imitation works as follows. Let m t be the number of miners currently present in the economy. At time t, agents mining on any colonized island j deliver a signal which is instantaneously spread in the system. A signal delivered from (x j,y j ) is received by a miner currently located at (x, y) 6= (x j,y j ), independently of all other delivered signals, with probability: w t (x j,y j ; x, y) = m t(x j,y j ) exp{ ρ[ x x j + y y j ]}, ρ 0. (6) m t We call w t (x j,y j ; x, y) the intensity of the signal. Notice that w t (x j,y j ; x, y) is increasing in the share of miners working on j and decays exponentially with the distance between source and receiver. Furthermore, each signal has a content c t (x j,y j ) equal to actual productivity of the island is emitted by: c t (x j,y j )= Q t(x j,y j ) m t (x j,y j ). (7) Agent i will simply choose the signal associated to the largest content among all signals she has received (and randomizing if ties occur). Let us suppose that the receiver i is a miner on j. If the selected technology h is not the one she is currently mastering (i.e. h 6= j), she will become an imitator (a i,t+1 = im ). She will then move toward the imitated island (one step per period) and following the shortest path leading to h. Therefore, she will adopt h after k = x h x j + y h y j time periods. This allows us to embody in the model the time-consuming nature exhibited by many processes of technological adoption and diffusion. Finally, once the imitated island is reached, she will turn again her state into miner, i.e. a i,t+k+1 = mi. If on the contrary she chooses to stay on her current island, nothing happens and she will keep working on j at time t Discussion Before describing the implementation of the model and discussing the results, some considerations are in order. First, in tune with the philosophy of agent-based and evolutionary modeling, we start by analyzing a very simple economy populated by naïve agents behaving according to routinary, myopic rules. For instance, the exploration rule (4) implies that agents are not aware of (and cannot learn) the fact that islands are on average more and more productive the further away one goes from the origin of the lattice, as the expected 8

10 location after k periods is simply the starting node: E[(x i,t+k,y i,t+k ) (x i,t,y i,t )] = (x i,t,y i,t ). Moreover, we make the extreme assumption that the activities of exploration, imitation and production are costless and mutually exclusive. In fact, miners cease to produce while imitating and exploring. This can be interpreted as a sort of opportunity cost agents must bear in order for diverting resources from production to R&D or imitation. In a more realistic picture, firms should have been endowed with additional decision rules governing allocation of resources among exploitation and exploration activities (see e.g. Nelson & Winter (1982)). The strategy of keeping as small as possible the microeconomics of firm behaviors allows us to focus on the effects of the purported engines of growth only (i.e. innovation, diffusion, etc.). Second, the parameters governing production, exploration, innovation and imitation define easily identifiable technological regimes. In particular, whether ɛ (0, 1] or ɛ =0 allows us to discriminate between economies in which endogenous innovation is permitted or not. Furthermore, α tunes the regime of returns to scale in production, with α > 1 meaning increasing returns to scale due e.g. to learning by doing or economies of agglomeration. In addition, λ and π tune the degree of notional opportunities in the economy. Indeed, a large λ lets average productivity of a newly discovered technology to be sensibly larger than that associated to currently known islands. Conversely, a smaller λ implies search processes characterized by small improvements upon currently mastered practices (i.e. incremental innovations). Likewise, a larger π induces a larger average number of per-period discoveries and thus is associated to economies where technological opportunities are very likely. Third, the strength of path-dependency in innovation depends positively on ϕ. Large ϕ 0 s mean that more skilled explorers (i.e. miners who have been more efficient in the past) are likely to discover more productive islands today and therefore to produce more in the future, thanks to a sort of learning-to-learn mechanism àlastiglitz (1987). Fourth, the process of knowledge diffusion governs the interaction regime in the model (see Fagiolo (1998) and Kirman (1998)). Indeed, the behaviors of any firm is directly affected by the information signals emanated by agents employing similar technologies. The parameter ρ 0 tunes the degree of locality of the interactions: the larger ρ, the more the process of diffusion of knowledge is local, since signals will tend to reach, in probability, only nearest neighbors. Two extreme cases are: (i) ρ =0, i.e. interactions are global, as information diffusion does not depend on the distance between source and receiver; and (ii) ρ =, i.e. no signals are spread and interactions are shut down. 9

11 3.5 Initial Conditions, Timing and Implementation Suppose that at time t =0aset of initial islands L 0 (together with their coordinates in the lattice) is given and that all agents are randomly distributed across the l 0 mines. Assume also that the intrinsic productivity coefficients of any initial island j L 0 is simply s(x j,y j )= x j + y j. In each t =1, 2,..., given current agents coordinates and states, the timing of decisions and events occurring in a generic iteration (i.e. in the time interval (t 1,t] ) runs as follows. First, agents take their decisions: miners update output and choose whether to start searching; explorers select the next portion of the lattice to explore (and, possibly, they find a new island); imitators keep approaching the technologies they have chosen to adopt. Second, interactions take place through information diffusion. Finally, all time-t system variables are accordingly updated and the next iteration starts. The model is an example of a so-called artificial economy (cf. Lane (1993a)andEpstein & Axtell(1996)). Unless the focus is not on particular stationary cases (e.g. ɛ =0), one is bound to analyze its main properties by resorting to computer simulations. Analytical solutions are not indeed achievable for the full-fledged form, because of the underlying complication of the stochastic processes updating micro and accordingly macro system variables. In the next sections we will present an overview of simulation results 4, with particular emphasis on the aggregate properties of the simulated time-series of the log of GDP, i.e. q(ω) ={log Q t,t=1,..., T ; ω}, whereω is a point in the parameter space Ω, thatis: ω Ω {(ρ, λ, α, ϕ, π, ɛ, N, T) < 2 + [1, ) [0, 1] 3 {1, 2,...} 2 } (8) To begin with, we will analyze how the model behaves in some benchmark parametrizations, in order to assess the role played by knowledge-specific increasing returns, imitation and exploration in the dynamics of the economy. In particular, we will start by addressing the question whether the model is able to display patterns of persistent growth and if so under which behavioral and system parametrizations (especially concerning the degree of open-endedness of the economy, as well as innovation and diffusion rates). 4 For a thorough discussion of the results presented in the following Sections and for extensions of the model, cf. Fagiolo (2000). 10

12 4 The Emergence of Self-Sustained Growth: Bounded vs. Open-Ended Economies A key feature of the model resides in its ability to allow for an endogenous evolution of the set of fundamentals of the economy. But, in the first place, what happens if one bounds, to some extent, the dynamics governing the progressive enlargement of the technological frontier? Put it differently, is the economy able to generate patterns of self-sustaining aggregate growth if one considers stationary environments where agents behave on the grounds of a fixed set of fundamentals? The answer to this question in no. Too see this, let us first analyzing the benchmark case of a bounded economy (i.e. one in which L t L, t) and considering two distinct setups. 4.1 Bounded Economies without Exploration Assume first no possibility of exploration whatsoever, i.e. ɛ =0. In this setup, agents can exchange information about a fixed set of technologies, but they cannot endogenously introduce innovations in the system. To study the behavior of the system, we can focus, without loss of generality, on economies composed of only two islands, i.e. L 0 = l 0 =2. In this case we may neglect any spatial consideration and suppose that the productivity coefficients (s 1,s 2 ) < 2 + also represent the technological distance between islands. More precisely, let (s 1,s 2 )=(1,s), s=1, 2,..., and suppose that if a miner working on island j {1, 2} at the beginning of time t s decides to imitate island j0 6= j, then she will reach j0 at the end of time t 1 and start producing at time t. Island 2 plays here the role of the best practice for s 2, while the case s =1depicts the benchmark case of homogeneous technologies. In either case, the dynamics of the economy is entirely driven by the process of information diffusion (cf. Section 3.3), until one out of the two technologies, say j, manages to capture all N agents. In that case, no signal can be emitted by the other island and therefore the economy locks-in at the steady state where total output is Q = s j N α. An example of the behavior of the time-series q t isshowninfigure(1a). As intuition suggests, however, path-dependency entailed by increasing returns will tend to drive all agents, through waves of imitation, toward the island characterized by the actual (not initial) highest productivity. This in turn implies non-ergodicity in the stochastic process governing output evolution and, consequently, potential inefficiency. More formally, define M jt as the random variable: number of agents mining on island j at time t, j {1, 2}. It can be easily shown (see Fagiolo (2000) for details), that, if 11

13 s 2, M t = {(M 1t,M 2t ),t 1} is a non-stationary, aperiodic Markov process with two absorbing states m + =(0,N) and m =(N,0). Moreover, let p s (m 0 ; α, ρ) be the absorption probability in island 1, i.e. the probability of being absorbed in the inefficient limit state if s 2 given m 0 = m 10 {1,...,N 1} and system parameters. Simulations indicate that p s (m 0 ; α, ρ) is non-increasing in s and ρ and non-decreasing in m 0 and α. Figures (1b) and (1c) show examples of the estimation of p s (m 0 ; α, ρ) for s =1, 2, as(m 0 ; α, ρ) vary in the relevant parameter space. Notice that when the initial number of inefficient adopters is below a certain threshold (which itself increases with the strength of returns to scale α and the technological gap s), the system will inevitably converge to the efficient outcome no matter how large are the incentives to stick to the initial choice. However, when m 0 goes through that threshold, the probability of ending up in the inefficient state becomes strictly positive and grows as the incentives to knowledge accumulation increase. In the limit, when only a few miners are initially aware of the superior technology and returns to scale are increasing (α >1), the probability that the system is absorbed by island 2 converges to zero. Finally, when s 2, themore information diffusion is local (i.e. the greater ρ), the smaller the average number of miners which leave their islands and, consequently, the less likely the event that waves of imitation triggers a migration from the efficient technique toward the inefficient one see Figure (1d). Therefore, for a given (m 0,α), the probability of being absorbed in island 1 will decrease with ρ (increasingly fast as s grows). 4.2 Bounded Economies with Exploration In a setup without exploration, non-ergodicity of the stochastic process M t implies that the long-run steady-state GDP level is determined by unpredictable, early waves of imitation (cf. David (1992)). As it happens in Polya urn schemes (cf. Arthur (1994)), the system locks-in in the long run. However, in Arthur s model lock-in occurs because population size increases without bound. This implies that the perturbations introduced by individual choices become irrelevant in the long-run. Conversely, in the reduced form of the model presented here the population size is constant and perturbations die away as soon as an island manages to capture all miners. In order to explore what happens when the perturbation rate does not vanish, we study a second benchmark setup where: (i) the probability of finding new islands (that is the innovation probability) is π =0as before; (ii) exploration (as well as information diffusion) is permitted (ɛ >0), but only inside the initial set of knowledge bases. In this economy miners can become explorers with some probability ɛ>0, but they will only be able to sail within the box containing all initial islands (or, equivalently, on a finite 12

14 regular lattice with periodic boundaries). This implies that, for a given population size, the lock-in of the system will not generally occur, since there is always a positive probability that non conformist decisions will induce phase-transitions in the system. Notice that here we allow for a high potential source of irrationality and idiosyncrasy in individual behaviors, because agents could always decide to leave the island they are working on, even though all agents are mining on it. In a two-islands setup, the economy is characterized as before by the Markov process M t, together with the stochastic process describing the current number of explorers. However, unlike the previous case, transition probabilities are not only influenced by the propensities to imitate technologies with higher revealed productivity, but also involve a certain probability of exploring. Islands represent here basins of attraction among which the system continually oscillates 5. The stochastic process of exploration/imitation yields persistent output fluctuations but only transitory growth. Over finite time periods, increasing returns and knowledge diffusion induce agents (on average) to move toward currently more efficient islands cf. Figures 2(a) and 2(b) for the two cases s 1 = s 2 and s 1 <s 2. However, exploration allows with positive probability de-locking bursts, also toward notionally less efficient islands. In a sense, persistent fluctuations are in this case generated by a problem of imperfect Schumpeterian coordination in presence of dynamic increasing returns to learning. 4.3 Exploring in an Open-Ended Economy: Some Qualitative Results In both stationary environments analyzed so far, self-sustaining growth emerges only if one superimposes an exogenous Solow-like drift on the best-practice production function. Otherwise, as long as agents behave on the grounds of fixed fundamentals, economic growth is a transient phenomenon. Consider now the more general case where ɛ > 0 and the economy is open-ended (i.e. agents explore in a technological space without boundaries). Since firms are able to endogenously induce a drift in the technological frontier, the economy exhibits, for a wide range of parameters, patterns of self-sustaining (exponential) growth, cf. Figure 3 6. In all these cases, many other interesting regularities do actually arise. Suppose to start 5 The properties of the stochastic process governing the evolution of the system are qualitatively similar to those discussed in Kirman (1993). For instance, when the ratio between willigness to explore and the size of the population (ɛ/n) decreases, the system tends to spend an increasing number of time periods close to the absorbing states of an ɛ =0economy. 6 All results reported in this section refer to the parametrization: π =0.1, ρ=0.1, ɛ=0.1, λ=1, ϕ = 0.5, N = 100,α = 1.5, T = Cf. Section (5) for an extensive Montecarlo investigation of the parameter space. 13

15 from a fairly uniform distribution of N agents working on the initial set of known islands L 0. First, the number of currently known islands linearly increases in time. However, both the percentage of known islands and the number of colonized technologies fall quickly and then follow a stationary process. This suggests that a typical evolution of the system runs as follows. In the first time periods, diffusion of knowledge drives agents to concentrate on a relatively small cluster of known islands which, thanks to dynamic increasing returns, tend to be the most efficient ones. Relatively ordered spatial patterns of colonized islands are then likely to emerge, due to the local nature of both the exploration and imitation processes. In Figure (4a), the path of expansion of a best practice proxy b t is plotted 7, together with four snapshots showing the locations of currently colonized islands in the positive orthant of the technological space for different time periods t =0, 500, 1000, While in the early time periods of the simulation small (stochastic) events select the region of the lattice where exploration will be initially carried through, the path-dependent nature of the overall process tends to keep the economy inside that region. Therefore, rare events (i.e. exceptional discoveries), feeding path-dependently upon diffusion and incremental innovations thereafter, might be able to trigger a self-reinforcing process whose ultimate outcome is a pattern of exponential growth. Indeed, some lucky explorers are likely to find intrinsically superior islands outside the realized economy. Although they might not be able to adequately exploit the opportunities of the new island by themselves, the extraordinary nature of their discovery might nevertheless induce other agents to move there in the future and, consequently, increase its actual productivity. This allows the system to avoid lock-in, provided that ɛ>0 and the technological regime is characterized by sufficiently strong opportunities (see Section 5 below). Second, in accordance with empirically observed patterns of innovation, diffusion and adoption (see e.g. Dosi (1982)), the model generates s-shaped diffusion curves in the number of agents currently mastering a given technology. Moreover, because many techniques are allowed to coexist over the same time intervals (if they exhibits sufficiently similar realized productivities), one usually detects overlapping diffusion patterns as those depicted in Figure (4b). As the set of current available technologies keeps enlarging due to the unceasing process of exploration and innovation, firms migrate toward more productive islands, entailing processes of diffusion which occur at different rates. These rates typically depend on the characteristics of the technologies involved in the process, the incentives provided by the economic environment and the features of the adopters themselves. In very general terms, the speed at which innovations are adopted (and replaced) is increasing in 7 We define b t =(x t,y t ), wherex t =max{ x jt,j L t } and y t =max{ y jt,j L t }, i.e. the vertex of the smallest rectangle containing all currently known islands whose distance from the origin is the maximum one. Notice that b t does not necessarily coincide with a known island. 14

16 both their absolute initial productivity distance and the extent to which interactions are global. Also, if information is diffused not too locally, radical innovations tend to retain their leadership much longer than incremental ones. Yet, the rate at which innovations are substituted is decreasing with the average willingness to explore of the agents in the system. 5 The Sources of Self-Sustaining Growth The basic conclusion stemming from the analyses presented so far is that patterns of exponential growth might be endogenously generated in the system only if firms are able to explore in an open-ended technological space. In this Section, we study by means of extensive Montecarlo (MC) exercises how system parameters affect the distribution of long-run average grow rates (AGR): g m (ω) = q m,t q m,0, (9) T where m =1,..., M is the MC run, T is the econometric sample-size, ω Ω is defined in eq. (8) and q m,t is the log of aggregate GDP at time t. In particular, we will ask how the overall performance of the economy, as measured by the mean of AGR: g M (ω) =M 1 M X m=1 g m (ω), (10) changes in different technological and learning regimes (i.e. in different regions of the parameter space) 8. A first clear-cut result that MC simulations point out is that everything else being constant g M (ω) appears to be positively influenced by: (a) the extent to which the system is fueled with innovation opportunities (i.e. larger λ and π); (b) the magnitude of path-dependency affecting the innovation process (i.e. larger ϕ); (c) the degree of globality of the information diffusion in the interaction process (i.e. smaller ρ). This claim is supported by the surfaces in Figures (5a) and (5b) where, for a given choice of α and ɛ, we plot MC mean of AGRs against (log 10 ρ, ϕ) in two distinct opportunity setups (i.e. for different values of π and λ). Notice that, as typically happens in 8 All results presented below are not affected by the particular choice of the AGRs. Indeed, employing alternative specifications as g 0 m =[(q m,t /q m,0 ) 1/(T +1) 1] or g 00 m =[(Q m,t /Q m,0 ) 1/(T +1) 1], will only change the scale of attainable growth rates. Moreover, we have chosen values of T in such a way that recursive Montecarlo mean and variance of AGR converge. Therefore, properties about g M (ω) are not influenced by the econometric sample size. Finally, in the chosen range for T and M, the Montecarlo variance of AGR is typically negligible. This allows us to avoid reporting confidence intervals for g M (ω). 15

17 evolving complex systems (see Batten (2000)), the causal relationships between system parameters and aggregate variables are characterized by threshold effects and non-linearities (see Figures (5c) and (5d)). On the one hand, path dependence linearly affects the mean of AGRs. On the other hand, as one gradually increases the rate of information diffusion, an abrupt change in AGRs usually arises around ρ (ω) = 1.0. Ifρ<ρ (ω), theperformance of the system is barely influenced by ρ. When ρ>ρ (ω), small changes in the degree of locality of interactions bring about dramatic consequences in the mean of AGRs. Let us turn now to study how the willingness to explore of the system (ɛ) affects AGRs. As intuition suggests, larger AGRs could be attained if the economy somehow manages to optimally solve the trade-off between exploitation and exploration (cf. March (1991) and Allen & McGlade (1986)). However, it turns out that the levels of willingness to explore required to optimally balance between exploitation and exploration strongly depend on the technological and learning regime which characterize the economy. As illustrated in Figures (6a-d), four distinct regimes emerge in setups where returns to scale are increasing (α >1). When no interaction takes place (ρ = ) and opportunities are low, higher exploration rates are totally harmful because agents hardly find radically new practices and, if they do, they cannot benefit from increasing returns to scale. Hence g M (ω) monotonically decreases with ɛ no matter the degrees of path-dependence (cf. Figure (6a)). Conversely, economies in which information is globally diffused (ρ =0) and innovators strongly benefit from learning by doing (high ϕ) typically maximize their AGR when all agents commit themselves to exploration and production on new islands only lasts one period (see Figure (6b)). Moreover, if information is spread locally i.e. 0 << ρ << as in Figure (6c) the overall performance of the economy increases either if few explorers are around or if there are very many: in the first case, a large population of miners can continually exploit both increasing returns to scale and incremental, path-dependent, innovations through smallscale migrations driven by local imitation. In the second case, thanks to local information diffusion, small clusters of colonized islands can immediate benefit from the large-scale introduction of innovations. The most interesting regime, however, arises in all other intermediate settings where MC mean of AGRs are maximized by an interior value of ɛ, cf. Figure (6d). The intuition here corresponds to that suggested in March (1991, p.71). As he points out, systems that engage in exploration to the exclusion of exploitation exhibit too many undeveloped new ideas and too little distinctive competencies, while, at the opposite extreme, they are likely to find themselves trapped in sub-optimal stable equilibria. In our model, this condition applies in two setups, namely: (a) agents face very large opportunities but they are unable to completely exploit dynamic increasing returns because information is not spread around; (b) interactions are global but knowledge does not accumulate as the 16

18 economy evolves. In both situations, higher economic performances cannot be attained by entirely committing either to technological search or to production. As a result, losses stemming from the exploitation-exploration trade-off are reduced by an appropriate balance between the two forces 9. This point arises even more strongly when one allows for heterogeneity in agents willingness to explore. Consider for instance an economy in which an initial distribution E =(ɛ 1,ɛ 2,...,ɛ N ), ɛ i [0, 1] and ɛ i 0 6= ɛ i 00 for some i 0 6= i 00 is given. To keep things simple, let us suppose that E is such that ɛ i = 0, i = 1, 2,...,bµNc and ɛ i = ɛ 0, i = bµnc +1,...,N,whereµ [0, 1] and ɛ 0 (0, 1]. The aggregate consequences of increasing µ s (in terms of economy s AGRs) are once again strictly related to the handling of the exploitation-exploration trade-off, which in turn depends on the prevailing technological and institutional regimes, cf. Figures (7a-c). Again, in all intermediate setups described above, AGRs are maximized by some 0 <µ (ω) < 1, withpoorperformances when the economy commits either small or too many resources in the exploration of unknown knowledge bases. 6 Growth Rates Volatility and System Performance Higher average economic performances are generated in the model if the economy is gradually injected by increasingly powerful sources of growth (i.e. stronger increasing returns to scale, more global knowledge diffusion, higher path-dependency and technological opportunities). It is then of interest to assess how the volatility of aggregate performances (both across MC samples and within time-series realizations) is affected by system parameters governing these forces. Despite what one could have expected, patterns of self-sustaining growth characterized by higher AGRs are not generally associated with overly increasing levels of growth rates volatility. On the one hand, a strong positive correlation emerges between g M (ω) and MC sample standard deviations: σ gm (ω) =[M 1 M X m=1 g 2 m(ω) g 2 M(ω)] 1 2, (11) so that the latter appear to be monotonically increasing with λ, π, ϕ and ρ, everything else 9 Simpler patterns arise when one analyzes how different regimes of returns to scale in production affect economic performance. When interactions are shut down (ρ = ), MC means of AGRs tend to be decreasing with α when opportunities are low (and only mildly increasing for large α s when they are high). Conversely, if information is globally diffused, AGRs are monotonically increasing in α for any (λ, π) and ϕ. 17

19 being constant 10. On the other hand, MC sample standard deviations never explode as one increases the strength of the sources of growth. Therefore, despite the self-reinforcing nature of the mechanisms triggering economic growth in the system (i.e. exploration, innovation and more efficient production), the model yields sufficiently ordered growth paths, which turn out neither to overlap nor to converge as long as one considers sets of GDP time-series generated by points in the parameter space far enough from each other. To illustrate this property, Figure 8 plots time-series describing the 5% and the 95% percentiles of the MC distributions q t (ω) ={q m,t (ω),m =1,...,M}, ast =1,...,T,in four different parameter setups (M = 10000). Notice that even in a global information / high opportunities setup, the band including the 90% of MC observations does not widen as T grows. Moreover, 90% confidence intervals do not overlap even for very small econometric sample sizes. Let us turn now to the properties of the within-sample volatility of growth rates timeseries (GRTS) h m (ω) ={h m,t (ω), t =1,...,T}, where: h m,t (ω) = q m,t q m,t 1. (12) q m,t 1 Here a first important result is that, unlike MC sample standard deviations, self-sustaining growth does not always imply a larger volatility in GRTS (for a given econometric sample size T ), as measured by the MC mean of its standard deviation: v u σ(h m (ω)) = t T 1 TX h 2 m,t(ω) [T 1 t=1 TX h m,t (ω)] 2. (13) In particular, when radical innovations are very likely, setups typically yielding self-sustaining growth (e.g. small ρ s, large ϕ s) are characterized by a lower magnitude of average volatility, whereas economies usually generating stationary GDP time-series or very mild growth display a higher GRTS variation cf. Figure (9a). Even more unexpectedly, persistently higher AGRs seem to be attained by the system through a process characterized by GRTS volatility decreasing in time (i.e. across subsequent phases of development). To illustrate this property, consider as done in Figure (9b) four prototypal environments yielding: (a) stationary GDP time-series; (b) levels of GDP evolving around a S-shaped trend; and self-sustaining growth emerging from (c) a low opportunities setup; or (d) a high opportunities setup. As one takes into account the time evolution of MC mean of the distributions of recursive standard deviation of GRTS (i.e. computed over enlarging econometric sub-samples {T 0,T 0 +1,...,T }, fort = T 0 +20, t=1 10 For a similar property displayed by actual time-series in a cross-section of countries, cf. Fatas (2000). 18

20 T 0 +21,...,T and T 0 =50) a striking pattern arises. Indeed, recursive standard deviation of GRTS appears to behave as T β 1, β>0, ineachoftheaboveenvironments. However, while in the stationary GDP case one has 1 β<2, as soon as some evidence of persistent growth emerges in the system, β becomes less than unity and recursive standard deviations turn out to be monotonically decreasing toward some positive constant level. In general, a negative correlation emerges between β and the overall performance of the economy: the more one fuels the system with opportunities and path-dependency, the higher the rate at which GRTS volatility, as measured by average recursive standard deviation, decreases in time. Therefore, the model seems to account for the appearance, over finite time periods, of distinct phases of development. Under structural conditions above certain thresholds, the economy displays an aggregate dynamics wherein phases of almost steady positive growth rates are punctuated by temporary slowdowns. Exponential growth thus emerges as the outcome of a process leading to ordered GDP time-series characterized by fairly moderate variability both across independent histories and, more importantly, within the sample path. 7 Statistical Properties of Simulated GDP Time-Series The foregoing exercises have attempted to shed some light on the mechanisms underlying the emergence of self-sustained growth in the model. In this Section we will ask whether (and if yes, under which technological regime) the model is able to generate simulated GDP time-series which display statistically properties similar to those empirically detected in actual output time-series (e.g. non-stationarity, auto-correlation in output growth, persistence of oscillations, etc.). Let us start to address this exercise in plausibility by noticing that (when they arise) patterns of self-sustaining growth are always associated in the model to difference stationary log(gdp) time-series (as opposed to trend-stationary ones). In fact, according to standard ADF tests and irrespective of the employed Dickey-Fuller regression specification onecannotrejectthenullofaunitroot(at5%), which, on the contrary, is systematically not accepted for both first differences q m,t andgrowthratesh m,t = Q m,t /Q 11 m,t 1. Even more interestingly, we find that the ways in which system parameters affect the likelihood of generating I(1) time-series (i.e. patterns of self-sustaining growth) are very similar to the ways in which system parameters affect system performances (i.e. mean of AGRs). Indeed, the behavior of MC mean of ADF(1) teststatisticst 1 (q m,t (w)) mimics the 11 For a critical discussion on trend vs. difference stationarity and drawbacks of ADF tests, cf. Fagiolo (2000). 19